Stochastická nelineárn. rní analýza betonových konstrukcí: : spolehlivost, vliv velikosti, inverzní analýza

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1 Stochastická nelineárn rní analýza betonových konstrukcí: : spolehlivost, vliv velikosti, inverzní analýza Drahomír Novák Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Brno Czech Republic

2 Outline Stochastic techniques for uncertainties simulation Software FReET Feasible Reliability Engineering Tool Development stimulations: Long-term focus of Brno reliability group (Vořechovský, Rusina, Lehký SARA project (Bergmeister, Pukl, Červenka, Strauss ) - To combine efficient methods of reliability and nonlinear analysis - Software ATENA+FReET=SARA - To provide an advanced tool for assessment of real behavior of concrete structures Selected types of applications (stochastic nonlinear analysis) 2

3 Stochastic techniques for uncertainties simulation Introduction computational demands Small-sample simulation of Monte Carlo type Imposing statistical correlation Simulation of random fields Sensitivity analysis Reliability analysis Inverse analysis 3

4 Two main categories of stochastic tasks/approaches Approaches focused on the calculation of statistical moments of response quantities (means, variances, etc.) response function Approaches aiming at the calculation of theoretical probability of failure limit state function R Z = g R ( x x x Kx ), 2,K i, = g ( ) z x, x2,k xi, Kxn p f = P Z ( 0) n 4

5 Reliability analysis - computational demands: Number of evaluation of limit state function Crude Monte Carlo Importance sampling Approximation FORM, SORM - design point calculation Response surface Cornell safety index, Curve fitting 0-00 MC LHS 5

6 Latin Hypercube Sampling The range (0; ) of PDF Φ(Y i ) of each random variable Y i is divided into N non-overlapping intervals of equal probability /N (McKay et al Iman & Conover 980, Iman & Shortencarier 984). The centroids are selected randomly based on random permutations of integers. Every interval of each variable is used only once during the simulation process. variable simulation 6

7 LHS: Step - simulation Huntington & Lyrintzis (998): Improvements to and limitations of LHS x ik, Sim ( ) b = N x f x dx a where a = F b = F k N k N Mean value: accurately Stand. deviation: significant improvement 7

LHS: Step 2 imposing statistaical correlation x x 2 variable y y 2 z z 2 Simulated annealing: Probability to escape from local minima Cooling - decreasing of system

8 LHS: Step 2 imposing statistaical correlation x x 2 variable y y 2 z z 2 Simulated annealing: Probability to escape from local minima Cooling - decreasing of system excitation Boltzmann PDF, energetic analogy simulation x 3 x 4 x 5 x 6 x 7 y 3 y 4 y 5 y 6 y 7 z 3 z4 z5 z6 z 7 kb ( ) P E e r ΔE T x 8 y 8 z 8 x NSim y NSim z NSim 8

9 Best ordering (all possible rank combinations). Is it possible to find the global minimum? simulation x NSim y NSim variable x y z x 2 y 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5 x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 2 z 8 z NSim One column remains stable. Others permute. There exist N Sim! ( ) N Var possibilities. In case of 6 simulations with 5 variables: ( ) 5 6! = possibilities.

excitation Boltzmann PDF, energetic analogy simulation x 3 x 4 x 5 x 6 x 7 y 3 y 4

10 LHS: Step 2 imposing statistaical correlation x x 2 variable y y 2 z z 2 Simulated annealing: Probability to escape from local minima Cooling - decreasing of system excitation Boltzmann PDF, energetic analogy simulation x 3 x 4 x 5 x 6 x 7 y 3 y 4 y 5 y 6 y 7 z 3 z4 z5 z6 z 7 kb ( ) P E e r ΔE T x 8 y 8 z 8 x NSim y NSim z NSim 0

11 Statistical correlation in LHS - optimization problem a i,j b i,j - the target correlation matrix - the actual correlation matrix min N, k ( a ) i, j bi, j i, j 2

12 Decrease temperature T Loops-times (~000) Change of ranks (offspring) Simulated annealing Improvement? No r = rand(0,) e = exp( E / T) r < e? Yes, accept changes Yes, accept changes No changes E k i = Err i Err i b = P r ( E ) e E k T b No 2

13 Numerical test : diminish spurious correlation 0 K= variables and 6 simulations.. ULHS, iterations (Spearman) 2. Simulated annealing, (PC 400MHz 3 sec ) 3

14 Diminish spurious correlation comparison Cholesky decomp. iterative ULHS (Spearman) E = overall Samples: Proposed algorithm (Simulated Annealing) E = 2 overall Difference:. < target correlation matrix <0.2 target correlation matrix <0.3 4

15 Numerical test imposition of target statistical correlation 0.2 K= variables and 6 simulations. Number of simulation - great influence. All possibilities 50 minutes Simulated annealing 3 sec (always finds local minima) 5

16 Simulated annealing - results Starting sampling matrix E = overall Samples: Proposed genetic algorithm (Simulated Annealing) E = 2 overall Difference:. < < <

17 variable LHS: Simulated annealing weighted 0,9 0,9 0,49 0,49 variable -0,9 optim -0,49 Resulting correlation matrix is positive definite and error is uniformly distributed among all coefficients - compromise Positive definiteness of K 7

18 variable LHS: Simulated annealing weighted 0,9 0,9 0,2 0,2 variable -0,9 optim 00-0,9 Resulting correlation matrix is positive definite and error is uniformly distributed among all coefficients Weighted method: suppression of selected coefficients Positive definiteness of K 8

19 Simulation of random fields Essential topic in stochastic continuum mechanics. The need for accurate representation and simulation in SFEM. Various methods... Orthogonal transformation of covariance matrix (Schuëller et al. 990, Liu et al. 995) Small number of random variables to represent random fields. Latin Hypercube Sampling (LHS) Small number of simulations. Combination: A new alternative method 9

20 T C XX = ΦΛΦ C = Λ YY Simulation - uncorrelated Gaussian random variables: Y T = [Y, Y 2,,, Y nr ]: Y MCS Simulation of random fields Orthogonal transformation of covariance matrix and LHS Φ - eigenvector matrix Λ - Cov. matrix in uncorrelated space (diagonal) : eigenvalues λ,λ 2,,λ nd X = ΦY Random field I longormal μ = 5.0 σ = LHS? Vořechovský, Novák - Icossar R aa ( ξ ) 2 ξ = σ 0 exp d sim. 6 sim

21 Comparison of convergence to target fields statistics Mean a) b) Standard 0.03 deviation c) 0.02 Dispersion d) 0.0 of Std Praha 64, Number of simulations n 0 Dispersion of mean LHS-mean- LHS-half- MCS- SCD SC SCD SC SCD SC LHS-mean- SCD SC LHS-half- MCS- SCD SC SCD SC

SFEM model with 3000 DOF random field to describe geometrical imperfections 500 random variables Novák, D., Lawanwisut, W., Bucher C. (2000).

22 Bucher, C. and Ebert, M. (2000) Load Carrying Behavior of Prestressed Bolted Steel Flanges Considering Random Geometrical Imperfections, PMC2000, University of Notre Dame, USA. SFEM model with 3000 DOF random field to describe geometrical imperfections 500 random variables Novák, D., Lawanwisut, W., Bucher C. (2000). Simulation of random field based on orthogonal transformation of covariance matrix and Latin Hypercube Sampling,, MC 2000, Monte Carlo. 500 reduction 28 Method Mean Value [MNm] Coeff. of Variation [%] LHS (32 samples) MCS (200 samples) Finite element model of flange Statistics of ultimate bending moment Realization of random field 22

23 Sensitivity analysis Nonparametric rank-order correlation between input variables ane output response variable Kendall tau τ = τ q ji,p j j = Spearman ( ),,2, K N i, r s = n( n )( n + ) Robust - uses only orders Additional result of LHS simulation, no extra effort Bigger correlation coefficient = high sensitivity Relative measure of sensitivity (-, ) 6 i= n d 2 i INPUT x, x,n INPUT q, q,n OUTPUT R R, N OUTPUT p p N 23

24 Reliability analysis Simplified rough estimates, as constrained by extremally small number of simulations (0-00)! Cornell safety index μ β = Z Curve fitting σ Z FORM, importance sampling response surface 24

25 The method of Hasofer and Lind, 974 Hasofer and Lind, important step Transformation of the limit state function into so-called standard space U = R μ R σ R S New variables with mean value 0 and standard deviation In the new coordinate system the line G=R-S no longer passes through origin HL safety index - the distance from the design point to origin correct in case of normally distributed variables, for nonnormally - a good approximation U 2 = S μs σ β = u Subject to g = 0 ( X ), T u p f = Φ( β ) G = R S = U + μ U R R 2 σ σ + μ ( ) ( ) S S 25

26 Identification of material parameters Numerical model of structure appropriate material model many material parameters Information about parameters experimental data recommended formulas engineering estimation 26

27 Identification of material parameters Primary calculation Correction of parameters: trial and error method sofisticated identification methods artificial neural network + stochastic calculations (LHS) 27

28 Artificial neural network Modeling of processes in brain (943 - McCulloch-Pitts Perceptron) Various fields of technical practice Neural network type Multi-layer perceptron: - set of neurons arranged in several layers - all neurons in one layer are connected with all neurons of the following layer 28

29 Artificial neural network NEURON: Output from neuron: y ( x) = f ( w p ) + b = f k k k k number of input impuls (,...,K) w k weight coefficient of connecting path from k-th neuron of previous layer p k impuls from k-th neuron previous layer b bias of neuronu f transfer function of neuron 29

30 Artificial neural network active period (simulation of Two phases: process) adaptive period (training) Training of network: - training set, i.e. ordered pair [p i, y i ] Minimization of criterion: input and output vector E = 2 N K ( v * y ) ik yik i= k = 2 N number of ordered pairs input - output in training set; required output value of k-th output neuron at i-th input; real output value (at same input). * y ik v y ik 30

31 Principle of identification method Structural response Material model parameters Stochastic calculation (LHS) training set for calibration of synaptic weights and biases 3

32 FReET software development Stand alone module - definition of reliability problem (user-defined limit state/response function) in programming language (C++, FORTRAN) DLL function or by equation interpretor Integration with software ATENA - nonlinear fracture mechanics of reinforced concrete structures (Červenka Consulting) SARA software shell 32

33 Software Freet Freet version. December 2004 Enhanced set of PDF a possibility to add new comparative values without need to perform simulation outputs organization and printing possibilities USB hasp Freet version.2 February 2005 Net version (BOKU computer lab installation) Hasp for SARA, ATENA, FREET sensitivity graphical output enhanced (also what-if-studies, parametric study) Freet version.3 June 2005 Weighting for correlation matrix input Response Surface basics File->New clearing results and input Graphics enhancement and checking Random fields basics Freet version.4 May 2006, new features: Graphics enhancement and checking Random fields implementation verified Possibility to define a parameter for easy parametric study with graphical output New type of probabilty distribution: Bounded normal PDF Automatic running of FREET from command line Freet version.5 January/February 2007, new features: More general interface to third-parties programs now DLL and BAT, EXE files communication via text input/output files FORM First Order Reliability Methods

34 FREET - Feasible Reliability Engineering Tool 34

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41 Input Xlimit.dll Output 4

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43 Z=R-E 43

$fracture mechanics analysis$

44 Stochastic NLFEM SARA Studio Probabilistic software FReET + Software for nonlinear fracture mechanics analysis ATENA 44

$post-peak behavior smeared crack approach crack band method fracture energy Crack band size: L ε = w L fixed or rotated cracks crack localization size-effect is$

45 Non-linear techniques and material models for concrete: : ATENA software Numerical core advanced nonlinear material models concrete in tension tensile cracks post-peak behavior smeared crack approach crack band method fracture energy Crack band size: L ε = w L fixed or rotated cracks crack localization size-effect is captured 45

environment visualization + interaction -2.43E+00-2.200E+00-2.000E+00 -.800E+00 -.600E+00 -.

46 Software ATENA Well-balanced approach for practical applications of advanced FEM in civil engineering Numerical core state-of-art background + user friendly Graphical user environment visualization + interaction -2.43E E E E E E E E E E E E E+00 M2: Reactions Component 2 46

47 Crack band method Numerical core advanced nonlinear material models concrete in tension tensile cracks post-peak behavior smeared crack approach crack band method fracture energy fixed or rotated cracks crack localization size-effect is captured 47

48 Run-time histogram of results

49 Software tools Software DLNNET: neural networks Software FReET: statistical, sensitivity and reliability analyses

50 Software communication for inverse analysis 50

51 Selected types of applications Example of FReET stand-alone application: Statistical analysis of concrete subway tunnel under Vltava river SARA - classes of tasks: Probabilistic analyses of concrete structures Statistical size effect studies Verification of (code) design formulas Identification of material model parameters (inverse analysis) 5

52 Large concrete subway tunnel under Vltava river in Prague (2002) Weight of tunnel Uplift force 2 random variables Imperfection of geometry, 4 segments Target: risk minimization Updating segments - convergence to required uplift force 52

53 Static scheme, forces acting on the tube A x x2 x2 B

54 Statistical simulation of uplift force 0,5 Uplift force [t/m]. 0-0,5 - -,5 Mean realizations 5% percentile 95% percentile Segments 54

55 Statistical simulation and measurement Uplift force [t/m]. 0,5 0-0,5 - -,5 Mean - simulation Reality 5% percentile 95% percentile Segments 55

56 Statistical simulation and measurement Uplift force [t/m]. 0,5 0-0,5 - -,5 Mean - simulation Reality 5% percentile 95% percentile Segments 56

57 Forces - barrels with water Number Absolute Frequency Histogram - [metro] [BremenoP][BremenoP] 0 t 64 t A x x2 x2 B

6 7 8 9 272 26 70 8 6 78 99 2 0 9 3 5 2 7 292 25 30

58 Probabilistic analyses of concrete structures: Box-girder prestressed bridge in Vienna 85E+0 Y X

59 Random variable description Symbol Units Mean value COV Distribution type Reference Concrete grade B500 6 Modulus of elasticity Ec GPa Lognormal Poisson's ratio μ Lognormal Estimation 6 Tensile strength ft MPa Weibull 6,7 Compressive strength fc MPa Lognormal 8 Specific fracture energy Gf N/m Weibull 6 Uniaxial compressive strain ε c Lognormal Reduction of strength cred Rectangular Estimation Critical comp displacement wd m Lognormal Estimation Specific material weight ρ MN/m Normal Prestressing strands 0 Modulus of elasticity Es GPa Lognormal 0 Yield stress fy MPa Lognormal 9 Prestressing force F MN Normal Area of strands As m Normal load [MN] deflection [m] Reliability index D D2 D3 D4 D5 D6 D7 D8 D mean Load [MN] COV of load = 0. COV of load = 0.2 Eurocode 59

Probabilistic analyses of concrete structures: Cantilever beam bridge in Italy 30 Safety index β 25 20 5 0 CoV = 0.5 CoV = 0.2 CoV = 0.0 CoV = 0.075 CoV = 0.

60 Probabilistic analyses of concrete structures: Cantilever beam bridge in Italy 30 Safety index β CoV = 0.5 CoV = 0.2 CoV = 0.0 CoV = CoV = 0.05 Eurocode Design Load Line Load [kn/m] Colle d Isarco bridge. Brennero highway, Italy Reliability index vs. load 60

61 Typical failures m Y 2.520E E E E E E m 3.420E E-04 X Shear m Y m X 6

62 Probabilistic analyses of concrete structures: Soil-structure interaction, spatial variability Stability of concrete tunnel tube in complicated geological conditions Influence of spatial variability of Young modulus and material constants of Drucker-Prager criterion (based on cohesion and angle of internal friction) Analyzed part 50 x 60m, diametr of tunnel m, wall thickness 0.5m Plain strain state, 5000 finite elements 5.657E E E E E E-0.55E-0.260E-0.365E-0.470E-0.575E-0.680E-0.78E-0 62

63 Four-point bending - different bending span Statistical size effect studies: Four Koide at al. Experiments on 4PB Statistical size effect! Cannot be captured at deterministic level 63

$Alt. I: No correlation between tensile strength and fracture energy Alt.$

64 Alt. I: No correlation between tensile strength and fracture energy Alt. II, III: High correlation between tensile strength and fracture energy 64

Four-point bending - different bending span Statistical size effect studies: Four 6 Nominal stress σn [MPa] 5 4 3 2 0 0 0. 0.2 0.3 0.

65 Four-point bending - different bending span Statistical size effect studies: Four 6 Nominal stress σn [MPa] Deflection [mm] Four-point bending and patterns of random fields Random load deflection curves (red curve deterministic calculation). 65

66 Dog-bone shaped concrete specimens in uniaxial tension Statistical size effect studies: Dog EXPERIMENT - Van Vliet and Van Mier, 997 Size A B C D E F D [mm]

67 Statistical size effect studies: Malpasset dam (failed 959) Calculation for different sizes, microplane M4 model Bažant, Vořechovský, Novák - Icossar

68 Size effect formulae and verification by statistical simulation 68

69 Verification of (code)) design formulas:shear failure of reinforced concrete beams BN00 BN50 BN25 BN2 0 Nominal strength [kn/m2] Experiments Statistical simulation ACI Eurocode 2 Collins Bažant 600 Nominal strength vs. size for different design alternatives: formulas, experiment and simulation Size (depth of beam) [m] 69

70 Failure surface approximmation Inverse analysis Training, stochastic preparation of training sets: classical Monte Carlo vs. Latin Hypercube Sampling methds 3 2 ( X) ax +bx + cx X d g + = a = 0,36355, b =,8046, c = , d = X / = Ha / m p, X 2 = Va m p 70

71 Inverse analysis a) Shape of failure surface: X X2 Aim: identification of parameters a,b,c,d Parametric study for small numbers of simulations 20,30,40 and 50 Same initial conditions (scattter of parameters, neural network type, same initiation of synaptic weights and biases to start training of network) 7

72 Inverse analysis a) X b) X X X2 Experiment Random realizations Experiment LHS - 30 simul. MC - 30 simul. LHS - 40 simul. MC - 40 simul error Training set (20 random realizations) resulting failure surfaces errors of identifications ,5 358,9 63,6 54,9 45,7 38, number of simulations LHS MC 5,5 0,5 72

73 Identification of material parameters: Shear wall test Reality Simulation Experiments by Maier and Thürliman,

74 Identification of material parameters: Shear wall test Variable Symbol Unit Mean value COV Modulus of elasticity E GPa Tensile strength ft MPa Compressive strength fc MPa Fracture energy GF N/m Compressive strain εc Max. comp. displacement wd m Bilinear diagram of steel for smeared reinforcement x fx x2 fx2 m kn m kn Randomization of material parameters preparation of training set

75 Identification of material parameters: Shear wall test FEM model in software ATENA 75

76 Identification of material parameters 20 random realizations training set 76

77 Identification of material parameters input 2 hidden layers output p 24 points on l-d diagram p 2 p y y 2 6 or 0 neurons in output layer linear transfer function y 6 or 0 p or 0 2 and 0 neurons in 2 hidden layers nonlinear transfer function 77

78 Identification of material parameters Spearman E f t f c G f ε c w d x fx x 2 fx 2 F 0,753 0,23 0,453 0,045-0,335-0,08-0,67 0,05-0,087-0,07 F 5 0,262 0,53 0,460 0,04-0,263-0,08-0,56 0,3-0,05 0,045 F 0 0,58 0,382 0,608 0,08-0,080-0,027-0,344 0,490 0,005 0,04 F max 0,29 0,34 0,636 0,054-0,042-0,053-0,307 0,537-0,009 0,7 Sensitivity of material model parameters: 6 parameters identified 0 parameters identified Parameters obtained from simulation of neural network: DLNNET E [MPa] f t [MPa] f c [MPa] G f [MN/m] e c [-] w d [m] x 6 par. 0 par. 29,9 33,0 2,47 2,47 34,5 35,3 75,0 77,85 2,5E-03 2,57E-03 3,00E-03 3,0E-03 2,72E-03 2,74E-03 fx 566,9 570,7 x 2,50E-02,47E-02 fx ,8 78

79 Identification of material parameters: Shear wall test Horizontal force [kn] S2 Experiment ATENA 6 parameters identified ATENA 0 parameters identified Horizontal displacement [mm] L-d diagrams obtained with identified parameters DLNNET E [MPa] f t [MPa] f c [MPa] G f [MN/m] e c [-] w d [m] x 6 par. 0 par. 29,9 33,0 2,47 2,47 34,5 35,3 75,0 77,85 2,5E-03 2,57E-03 3,00E-03 3,0E-03 2,72E-03 2,74E-03 fx 566,9 570,7 x 2,50E-02,47E-02 fx ,8 79

80 Experimental works (VUSTAH) 3-point bending experiment of fibrereinforced concrete notched beams, Specimen parameter Units Value Length of specimen Width of specimen Depth of specimen Depth of notch Weight Span mm mm mm mm kg mm ,

81 Experimental works (VUSTAH) 9 experimental load-deflection curves experiment Load [N] Deflection [mm] 8

$Consulting) Material model SBETA nonlinear fracture mechanics: - smeared cracks model$ $fracture energy - softening model for fibre-reinforced concrete (parameters c and c 2$

82 Virtual numerical simulation of experiment Nonlinear analysis software ATENA (Červenka Consulting) Material model SBETA nonlinear fracture mechanics: - smeared cracks model (fixed or rotated cracks) - crack band method (localization limiter) - crack opening fracture energy - softening model for fibre-reinforced concrete (parameters c and c 2 ) 82

83 Inverse analysis Experimental + virtual (numerical) load-deflection curves experiment simulation Load [N] Deflection [mm] 83

84 Statistics of fracture-mechanical parameters Inverse analysis based on neural networks Parameter Unit Mean value Standard deviation Coefficient of variation in % Maximum failureload kn,49 0,36 24, Deflection a maximum load mm 0,67 0,20 29,3 Moduus of elasticity GPa 5,4,68 30,9 Tensile strength MPa,3 3,39 29,9 Fracture energy J/m ,5 Softening parametr c - 0,9 0,02 2,5 Softening parametr c 2-0,

85 Conclusions Methods for statistical, sensitivity and reliability analyses, suitable for analysis of computationally intensive problems (eg. continuum mechanics, FEM) Software tools FREET and SARA - for the assessment of real behavior of concrete structures, can be applied for any problem of quasibrittle modeling of concrete structures A wide range of applicability both practical and theoretical - gives an opportunity for further intensive development of both methods and software 85

Uncertainty modelling using software FReET

Uncertainty modelling using software FReET D. Novak, M. Vorechovsky, R. Rusina Brno University of Technology Brno, Czech Republic 1/30 Outline Introduction Methods and main features Software FReET Selected